Any "good" cooler should not need ice more than once a day (of course, that assumes your cooler is in the cool, shaded, resort room, not the hot, sunny, car).
If you can, freeze all the "liquid" items you plan to take with (bottled water, milk, orange juice, etc.). Refrigerate everything else before you leave so it is cold before you pack the cooler.
I also find it helps to cool the cooler first. A few hours before you pack the cooler, fill it with cold water and some ice to get the inside cool. It will suck up less of the cold when you start packing.
We also take along a number of heavy-duty/freeze "ziplock-type" bags. Fill these with the ice and put in the cooler. Keeps the water contained and help avoid soggies.
Also, air space is not your friend. IF your cooler is NOT packed full, it will heat up faster because of the extra air space. Take along some extra towels, and lay them on the top of the food/stuff in your cooler to fill up any extra space.
I don't make any guarantees of having frozen water still at the end of the week, but you may make it half way. It depends on how often you are in the cooler too. If you take the above steps, you will certainly be able to keep stuff pretty cold for the week.